Properties

Label 2-7600-1.1-c1-0-154
Degree $2$
Conductor $7600$
Sign $-1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s + 1.56·7-s − 0.561·9-s − 4·11-s + 6.68·13-s − 7.56·17-s + 19-s + 2.43·21-s − 4.68·23-s − 5.56·27-s + 6.68·29-s − 3.12·31-s − 6.24·33-s + 6·37-s + 10.4·39-s − 4.24·41-s − 11.1·43-s − 10.2·47-s − 4.56·49-s − 11.8·51-s + 0.438·53-s + 1.56·57-s + 1.56·59-s + 2.87·61-s − 0.876·63-s + 1.56·67-s − 7.31·69-s + ⋯
L(s)  = 1  + 0.901·3-s + 0.590·7-s − 0.187·9-s − 1.20·11-s + 1.85·13-s − 1.83·17-s + 0.229·19-s + 0.532·21-s − 0.976·23-s − 1.07·27-s + 1.24·29-s − 0.560·31-s − 1.08·33-s + 0.986·37-s + 1.67·39-s − 0.663·41-s − 1.69·43-s − 1.49·47-s − 0.651·49-s − 1.65·51-s + 0.0602·53-s + 0.206·57-s + 0.203·59-s + 0.368·61-s − 0.110·63-s + 0.190·67-s − 0.880·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 1.56T + 3T^{2} \)
7 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 6.68T + 13T^{2} \)
17 \( 1 + 7.56T + 17T^{2} \)
23 \( 1 + 4.68T + 23T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + 3.12T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 0.438T + 53T^{2} \)
59 \( 1 - 1.56T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 - 1.56T + 67T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + 3.12T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.979194033709205134588573104057, −6.77676453572229197921407598262, −6.25143737185503514656713328076, −5.37551063084150066520473559135, −4.63260863803923493618589361226, −3.82128941654395237845097655473, −3.08095165122504694278971159328, −2.28722451329898414776339778583, −1.53809868931927291377650552111, 0, 1.53809868931927291377650552111, 2.28722451329898414776339778583, 3.08095165122504694278971159328, 3.82128941654395237845097655473, 4.63260863803923493618589361226, 5.37551063084150066520473559135, 6.25143737185503514656713328076, 6.77676453572229197921407598262, 7.979194033709205134588573104057

Graph of the $Z$-function along the critical line